Definition

The cokernel of this inclusion, hence the quotientC•(X)/C•(A)C_\bullet(X)/C_\bullet(A) of C•(X)C_\bullet(X) by the image of C•(A)C_\bullet(A) under the inclusion, is the chain complex of AA-relative singular chains.

A boundary in this quotient is called an AA-relative singular boundary,

Remark

This means that a singular (n+1)(n+1)-chain c∈Cn+1(X)c \in C_{n+1}(X) is an AA-relative cycle if its boundary∂c∈Cn(X)\partial c \in C_{n}(X) is, while not necessarily 0, contained in the nn-chains of AA: ∂c∈Cn(A)↪Cn(X)\partial c \in C_n(A) \hookrightarrow C_n(X). So it vanishes only “up to contributions coming from AA”.

Properties

Long exact sequences

Proposition

Let A↪iXA \stackrel{i}{\hookrightarrow} X. The corresponding relative homology sits in a long exact sequence of the form

Proof

This is the homology long exact sequence induced by the given short exact sequence0→C•(A)↪iC•(X)→coker(i)≃C•(X)/C•(A)→00 \to C_\bullet(A) \stackrel{i}{\hookrightarrow} C_\bullet(X) \to coker(i) \simeq C_\bullet(X)/C_\bullet(A) \to 0 of chain complexes.

Proposition

Let B↪A↪XB \hookrightarrow A \hookrightarrow X be a sequence of two inclusions. Then there is a long exact sequence of relative homology groups of the form

Excision

Let Z↪A↪XZ \hookrightarrow A \hookrightarrow X be a sequence of topological subspace inclusions such that the closureZ¯\bar Z of ZZ is still contained in the interiorA∘A^\circ of AA: Z¯↪A∘\bar Z \hookrightarrow A^\circ.

Here the left horizontal morphisms are the above isomorphims induced from the deformation retract. The right horizontal morphisms are isomorphisms by prop. and the right vertical morphism is an isomorphism since it is induced by a homeomorphism. Hence the left vertical morphism is an isomorphism (2-out-of-3 for isomorphisms).

Proof

This is the special case of prop. for AA a point.

Examples

Basic examples

Example

The reduced singular homology of the nn-sphereSnS^{n} equals the Sn−1S^{n-1}-relative homology of the nn-disk with respect to the canonical boundary inclusion Sn−1↪DnS^{n-1} \hookrightarrow D^n: for all n∈ℕn \in \mathbb{N}

Proof

The inclusion Xk−1↪XkX_{k-1} \hookrightarrow X_k is clearly a good pair in the sense of def. . The quotient Xk/Xk−1X_k/X_{k-1} is by definition of CW-complexes a wedge sum of kk-spheres, one for each element in kCellkCell. Therefore by prop. we have an isomorphism Hn(Xk,Xk−1)≃H˜n(Xk/Xk−1)H_n(X_k , X_{k-1}) \simeq \tilde H_n( X_k / X_{k-1}) with the reduced homology of this wedge sum. The statement then follows by the respect of reduced homology for wedge sums as discussed at Reduced homology - Respect for wedge sums.